
We note that the norm constraint here has dimension \(c/B\)  and \(\mathbf {w}_{1b}\)  has dimension \(sd/B\) , so by [1]}, [2]} this strong duality result from Lemma REF  requires that \(m^* \le 1/B\min \lbrace sd, c\rbrace \) . This is equivalent to
\(p_{BFN}^* &= \max _{\mathbf {V}_i} -\sum _{i=1}^n \mathcal {L}^*(\mathbf {V}_i, \mathbf {Y}_i) \nonumber \\\mathrm {s.t.}~& \Vert \sum _{i=1}^n {\mathbf {V}_i^{(b)}}^\top \mathrm {circ}({\mathbf {X}}_i^{(b)}) \Vert _2 \le \beta ~\forall b \in [B]\) 
